Bounded convergence theorem for abstract Kurzweil-Stieltjes integral

TL;DR

This paper establishes a proof of the Bounded Convergence Theorem for the abstract Kurzweil-Stieltjes integral, extending classical convergence results to Banach space-valued functions with an elementary approach.

Contribution

It provides the first elementary proof of the Bounded Convergence Theorem for the abstract Kurzweil-Stieltjes integral applicable to Banach space-valued functions.

Findings

The theorem holds for Banach space-valued functions.

The proof is simplified and does not rely on the Arzela Lemma.

It extends classical convergence theorems to a broader integral setting.

Abstract

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann-Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzela or Arzela-Osgood or Osgood Theorem. In the setting of the Kurzweil-Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the Young-Stieltjes integral by Hildebrandt in his monograph from 1963. However, it is clear that the proof by Hildebrandt cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzela Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil-Stieltjes integral in a setting elementary as much as possible.